1. Introduction

Large-eddy simulation (LES) is a powerful tool for predicting and evaluating thermal and wind environments in urban areas. LES reproduces thermals, gusts, and pollutant dispersions more accurately than the conventional Reynolds-averaged Navier–Stokes (RANS) model because LES directly resolves gridscale turbulence. However, inflow boundary conditions with turbulent components must be provided to fully utilize this feature. The simplest method to achieve this is to impose periodic boundary conditions on the lateral (inflow–outflow) boundaries (e.g., Walton and Cheng 2002; Kempf et al. 2005). However, this method assumes that identical surface conditions persist. Therefore, the experimental setup was limited to a simple setup. This is a well-known problem in weather simulations of real cities.

In the computational fluid dynamics (CFD) field, various methods for generating turbulent components at the inflow boundary have been developed to overcome this problem, and their simulation accuracies have been investigated (e.g., Keating et al. 2004; Tabor and Baba-Ahmadi 2010; Wu 2017; Vasaturo et al. 2018; Plischka et al. 2022). The methods used to generate the turbulent components can be classified into precursor and synthetic turbulence methods.

The precursor method generates a turbulent component by presimulating the flow under the desired conditions. Examples of precursor methods include the full-size presimulation method for creating a database of inflow turbulence (e.g., Bou-Zeid et al. 2009; Yoshida et al. 2018) and the recycle–rescale (R-R) method (Lund et al. 1998; Kataoka and Mizuno 2002). In the R-R method, the turbulent component is recycled downwind to the inflow plane to generate turbulent components. The R-R method was designed to generate turbulence without considering atmospheric stratification.

In the synthetic turbulence method, the turbulent component is artificially generated. Examples of synthetic turbulence methods include methods based on the Fourier transform (e.g., Lee et al. 1992; Kondo et al. 1997), synthetic eddy method (e.g., Jarrin et al. 2006; Poletto et al. 2013), random flow generation method (e.g., Huang et al. 2010), turbulence spot method (e.g., Kröger and Kornev 2018), and a method based on the combination of the Cholesky decomposition of Reynolds stress and digital filters (e.g., Klein et al. 2003; Xie and Castro 2008). The generated turbulent component satisfied the desired turbulence statistics such as the spectrum, spatiotemporal correlation, and Reynolds stress. In particular, the presimulation can be omitted when the turbulence statistics are obtained in advance. This indicates that synthetic turbulence methods can generate turbulent components at a lower computational load than precursor methods.

Most inflow turbulence generation methods exclude the effects of atmospheric stratification in simulations. Several methods that consider atmospheric stratification have recently been proposed in the field of CFD. The first approach was to modify the standard R-R method. Nakayama et al. (2008) applied a modified R-R method to an unstable atmosphere by considering temperature fluctuations. However, the temperature fluctuations were not recycled and were considered passive scalars. Jiang et al. (2012) adopted a similar approach. Their results suggested that these methods could be useful for simulations under nonneutral atmospheric conditions. However, problems remain when the R-R method is applied to thermally driven convective boundary layer (CBL) simulations. A problem with the modified R-R method is the scaling of velocity and potential temperature. The scaling parameters in the standard and modified R-R methods were based on turbulent boundary layer properties (i.e., mechanical mixing). The scaling parameters should be based on the CBL properties (i.e., thermal properties) to apply the R-R methods to CBL simulations. An additional problem with the modified R-R method is that convective vortices, such as thermals, have not been reproduced in the driver region. The turbulence field, including thermals, should be recycled when considering thermally driven CBLs.

The second approach is to extend the standard digital filter–based (DF) method. Okaze and Mochida (2017a) extended the standard DF method using a 3 × 3 Reynolds stress of momentum to 4 × 4 Reynolds stresses, including scalar stress (hereafter referred to as the extended DF method). Okaze and Mochida (2017b) considered that this scalar was temperature and performed an LES of the flow around a building. Their findings were consistent with wind tunnel experiments conducted under similar conditions. Sessa et al. (2020) validated this method under a stable atmosphere. However, the applicability of this method to the growth of thermally driven CBLs should be evaluated.

Meteorological LES occasionally targets both buoyancy turbulence and time-varying lateral boundaries. The cell perturbation method (CPM; Muñoz-Esparza et al. 2014) has been proposed for applications in such cases. This method added potential temperature perturbations near the upwind boundary to trigger turbulence. Because users do not need to specify the inflow boundary precisely, the CPM is suitable for connecting mesoscale meteorological models (RANS models) and microscale meteorological models (LES models), where the boundary values vary temporally and spatially. Muñoz-Esparza et al. (2015) compared the performance of the CPM and standard DF. They observed that the CPM could produce turbulence in a shorter driver region than the DF. Lee et al. (2019) conducted an LES using the CPM for an actual city. They recommended using a turbulence generation method, such as the CPM, because buildings inside the computational domain alone cannot generate sufficient turbulence.

In this study, we applied inflow turbulence generation methods developed in CFD fields to meteorological simulations of thermally driven CBLs. The primary objective of this study was to extend the R-R method to simulations of thermally driven CBLs. Moreover, this study investigated whether the extended DF method could be applied to simulations in which thermally driven CBLs are growing. The methods discussed in this study could be new alternatives to the inflow turbulence generation method, which is expected to be used for downscaling from meteorological mesoscale to microscale LES models.

2. Method

Simulations were performed for CBLs growing in an idealized city under constant spatiotemporal mean inflow. Under these conditions, we investigated the variations in wind speed, potential temperature, and turbulence [gridscale (GS) turbulent kinetic energy (TKE) and heat fluxes] with boundary conditions.

b. Computational domain and initial condition

The horizontal and vertical resolutions were 40 m each. The number of grid points in the east–west, north–south, and vertical directions was set to (nx, ny, nz) = (412, 256, 62). Time integration was performed for 2 h for all the boundary conditions. Analysis was performed using the results from the last 10 min. The details of the boundary conditions are described in section 2c. The initial wind speed was 5 m s⁻¹ (westerly wind) above the 200-m height. Below 200 m, the wind speed weakened in a logarithmic manner. A uniform random number at a magnitude of O(10⁻³) m s⁻¹ was added to the initial flow as the initial gridscale turbulence.

When using the R-R method described in section 2c, the driver region was added to the main domain. The size of the driver region was nx = 100 grid points (Fig. 1). Thus, the total number of grid points becomes (nx, ny, nz) = (512, 256, 62) when the R-R method is used.

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Schematic of the computational domain with the driver region for the R-R method.

Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0053.1

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An outflow boundary condition was used. The lateral boundary parallel to the main flow was a periodic boundary condition.

For the bottom boundary condition of momentum, the roughness parameter is given as z₀ = 1.0 m. The momentum flux at the bottom boundary is determined using the bulk method. The bulk transfer coefficient of momentum flux was calculated based on the method described by Mascart (1995). A sensible heat flux of 100 W m⁻² is provided.

Under the top boundary condition, a free-slip condition was imposed on the momentum with a damping layer above an altitude of 1500 m to prevent the reflection of gravity waves. The top potential temperature boundary was fixed at the initial potential temperature.

c. Inflow boundary condition

For the inflow boundary condition, three methods were implemented in addition to the reference periodic boundary condition (PER): the original method of Cholesky decomposition of the Reynolds stress and digital filter (original DF), extended method of Cholesky decomposition of the Reynolds stress and digital filter (extended DF), and extended R-R method. An extension of the R-R method is to enable it to deal with a thermally driven CBL (without considering the recycling scalar in the original R-R). To achieve this, we rescaled and recycled the wind speed and potential temperature perturbations using thermally driven CBL parameters. An extension of the DF method was proposed by Okaze et al. (2017a) to generate the turbulence components of wind speed and scalar using a 4 × 4 Reynolds stress, including scalars. As this method has not been applied to thermally driven CBL simulations, we investigated its performance. The major improvement in both methods is that they can generate turbulence components, including the potential temperature. The original DF, which did not consider potential temperature perturbations, was used as a baseline for the performance evaluation of the extension to consider potential temperature perturbations. In contrast, the original R-R cannot be used as a baseline method for evaluation because it does not consider the potential temperature. Therefore, only the original DF was used as the baseline method in this study. Note that the proposed methods can only be applied to cases where the flow enters from one side and exits from the opposite side. The mean wind speed was fixed at the initial conditions.

In the original and extended DF methods, the mean profiles of the potential temperature and Reynolds stress are required in addition to the vertical profile of the mean wind speed. The potential temperature and Reynolds stress profiles were obtained from calculations using the reference PER case. A background pressure gradient was used to maintain wind speed in the PER calculation.

1) Extended R-R method

This method is an extension of that proposed by Lund et al. (1998), which was originally developed for neutral atmospheres. The main objective of this method is to recycle the turbulent components. A recycling plane was placed in the region, and the turbulent components in the plane were recycled back to the inflow boundary to rapidly generate turbulence. Lund et al. (1998) divided the atmosphere into two layers: the turbulent boundary layer and the velocity deficit layer. The thickness of the turbulent boundary layer and friction velocity were used as the scaling parameters for the turbulence. However, in the case of CBL, both scale parameters were considered inappropriate. The height of the CBL was more suitable for the length scale than the thickness of the turbulent boundary layer. This is because the height of the CBL is on the scale of larger eddies (containing the highest energy). In such cases, the vertical velocity scale in the CBL is more suitable than the friction velocity as a scaling parameter for the wind speed. Therefore, the following length scale D (Stull 1988) and velocity scale w^* (Deardorff 1970) are used in this study:

$D={\left(\frac{2}{c_p{\rho }_a\text{Γ}}{\displaystyle {\int }_0^{\tau }Hd\tau }\right)}^{1/2},\text{\hspace{1em}and}$(1)

$w^{\text{*}}={\left(\frac{g}{\text{Θ}}DH\right)}^{1/3}.$(2)

Using these scale parameters, the velocity and potential temperature are scaled as follows:

${{\mathbf{u}}^{″}}_{\text{inlt}}^{\text{inner}}=\frac{w_{\text{inlt}}^{\text{*}}}{w_{\text{recy}}^{\text{*}}}{{\mathbf{u}}^{″}}_{\text{recy}}^{\text{inner}}=\sqrt[3]{\frac{D_{\text{inlt}}}{D_{\text{recy}}}}{{\mathbf{u}}^{″}}_{\text{recy}}^{\text{inner}}\hspace{0.17em}({\mathit{z}}_{\text{inlt}}^{\text{*inner}}),\text{\hspace{1em}and}$(3)

${{\theta }^{\prime }}_{\text{inlt}}^{\text{inner}}=\frac{D_{\text{inlt}}}{D_{\text{recy}}}{{\theta }^{\prime }}_{\text{recy}}^{\text{inner}}\hspace{0.17em}(z_{\text{inlt}}^{\text{*inner}}).$(4)

Here, “inner” represents the value inside the CBL; “inlt” and “recy” indicate the inlet and recycle planes, respectively; z^* denotes the dimensionless height; u″ represents the deviation from the spanwise mean vertical profile in the inlet/recycle plane [${\mathbf{u}}^{″}=\mathbf{u}-\overline{\mathbf{u}}(\mathbf{z})$]; and θ′ represents the deviation from the reference potential temperature Θ. At the inflow boundary, the vertical profile of the wind speed, similar to the initial condition, was fixed as an average profile, and recycled perturbation was added to these profiles. Owing to recycling the potential temperature, in addition to the wind speed, the preparation of the mean profile of the potential temperature can be eliminated. When scaling the potential temperature, the potential temperature increase was assumed to be linear with an increase in CBL height.

The heights of the CBL in the inlet and recycling planes were estimated using the following equations:

$D_{\text{inlt}}={\left(\frac{2}{c_p{\rho }_a\text{Γ}}{\displaystyle {\int }_0^{\tau }Hd\tau }\right)}^{1/2},\text{\hspace{1em}and}$(5)

$D_{\text{recy}}={\left(\frac{2}{c_p{\rho }_a\text{Γ}}{\displaystyle {\int }_0^{\tau +x/u_{\infty }}Hd\tau }\right)}^{1/2},$(6)

where x denotes the length of the driver region (4 km in this study).

In the upper atmosphere, the turbulence component is set to zero to maintain the initial state and suppress the generation of spurious gravity waves. The equations used are as follows:

${u^{\prime }}_{\text{inlt}}^{\text{outer}}=0,\text{\hspace{1em}and}$(7)

${{\theta }^{\prime }}_{\text{inlt}}^{\text{outer}}=\text{Γ}z.$(8)

Lund et al. (1998) did not recycle the scalar values. Mayor et al. (2002) extended this method to land–sea breezes. However, scaling was not performed because the boundary layer height was defined by the inversion layer in the driver region. Nakayama et al. (2008) extended the method to simulate thermally stratified wind tunnels. However, the effect of buoyancy was not considered in the driver region, and temperature scaling was not performed. Therefore, the thermals were not included in the generated turbulence. In addition, the temperature profile did not change with time. The extended R-R method performs scaling of both the wind speed and potential temperature to overcome these problems.

2) Extended method of Cholesky decomposition of Reynolds stress and digital filter (extended DF)

Okaze and Mochida (2017a) proposed this method. This method generates turbulent components at the inflow boundary from the Cholesky decomposition of a 4 × 4 Reynolds stress profile. Following Okaze and Mochida (2017a, 2018), wind speed and potential temperature f_i is expressed using the time-averaged value 〈f_i⟩ and fluctuation $f_i^{\prime }$ [Eq. (9)]:

$f_i=⟨f_i⟩+f_i^{\prime }.$(9)

The fluctuation $f_i^{\prime }$ can be expressed using the Cholesky decomposition of Reynolds stress a_ij:

$f_i^{\prime }=a_{ij}{\text{Ψ}}_j.$(10)

Here, Ψ_j satisfies 〈Ψ_j⟩ = 0, 〈Ψ_iΨ_j⟩ = δ_ij. The Ψ_j is given to satisfy the following time and space correlation using an integral time scale (T) and length scale (L):

$\frac{⟨{\text{Ψ}}_j(t){\text{Ψ}}_j(t+\tau )⟩}{⟨{\text{Ψ}}_j(t){\text{Ψ}}_j(t)⟩}=\exp \left(-\frac{\tau }{T}\right),\hspace{0.17em}\frac{⟨{\text{Ψ}}_j(r){\text{Ψ}}_j(r+\lambda )⟩}{⟨{\text{Ψ}}_j(r){\text{Ψ}}_j(r)⟩}=\exp \left(-\frac{\lambda }{L}\right).$(11)

The integral length scale L was calculated based on the study by Nakanishi and Niino (2009), using the results of the calculations with the reference PER as follows:

$L=0.23\frac{{{\displaystyle \int }}_0^{\infty }q\mathit{zdz}}{{{\displaystyle \int }}_0^{\infty }qdz}.$(12)

Here, $q=\sqrt{2k},\hspace{0.17em}k$ is the turbulent kinetic energy. The time scale was calculated from Taylor’s frozen turbulence hypothesis using the length scale and mean inflow wind speed. The time advance of Ψ_j at the point (m, n) on the inflow plane at time t [written in Ψ_j(m, n, t)] is expressed as follows:

${\text{Ψ}}_j(m,n,t+\text{Δ}t)={\text{Ψ}}_j(m,n,t)\exp \left(-\frac{\text{Δt}}{T}\right)+{\psi }_j(m,n,t+\text{Δ}t){\left[1-\exp \left(-\frac{2\text{Δ}t}{T}\right)\right]}^{1/2}.$(13)

Here, ψ_j(m, n, t) is expressed as follows:

${\psi }_j(m,n,t)={\displaystyle \sum _{m^{\prime }=-N_y}^{N_y}{\displaystyle \sum _{m^{\prime }=-N_z}^{N_z}{b}_{m^{\prime }}{b}_{n^{\prime }}{r}_{m+m^{\prime },\hspace{0.17em}n+n^{\prime }}}}.$(14)

The random number r_i satisfies 〈r_i⟩ = 0, 〈r_ir_j⟩ = δ_ij. The term b_k is a filter coefficient and is expressed as follows:

$b_k=\frac{\widetilde{b_k}}{{\left({\displaystyle \sum _{j=-N}^N\widetilde{b_j^2}}\right)}^{1/2}},\hspace{0.17em}\widetilde{b_k}=\exp \left(-\frac{\text{Δ}x|k|}{L}\right).$(15)

In this study, we imposed time-varying mean profiles of scalar and Reynolds stresses. The mean wind speed profiles were fixed to the initial wind profile. The average profile and spatiotemporal average profile of the turbulence were obtained every 10 min from the PER calculation. Vertical profiles are shown as representative values for 10 min (Fig. 2). Turbulent components were generated based on these 10-min values. This provides a time-varying inflow boundary that satisfies the mean and turbulence information obtained from the calculations using the PER.

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Schematic of the extended DF method. The blue arrow indicates the time evolution of the model, and the orange arrow indicates the time evolution of turbulence generation.

Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0053.1

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3) Original method of Cholesky decomposition of Reynolds stress and digital filter (original DF)

This method is the original version of the extended DF method described in (2). This method does not generate a turbulence component for the potential temperature. To achieve this, all Reynolds stress components related to the potential temperature are set to zero [similar to the method of the “case-mean” in Okaze and Mochida (2017a)]. The mean values at the inflow boundary are given in the same manner as in the extended DF described in (2).

Table 1 summarizes the information required to use each inflow turbulence generation method. In the original and extended DF methods, the mean vertical profiles of the wind speed, potential temperature, integral length scale, and Reynolds stress are mandatory to generate the turbulence components. By contrast, the extended R-R requires only the mean vertical profiles of the wind speed. CBL height is optional information to make the accuracy of the CBL height better.

Table 1.

Information needed to use inflow turbulence generation methods in this study.

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3. Result and discussion

Figure 3 depicts the vertical profiles of wind speed in the main flow direction at each downwind location. The wind speed in the case of the original and extended DF is at a deficit at 2 and 4 km downwind and becomes similar to that of the PER at 8 km downwind. The extended R-R method showed vertical profiles resembling those in the PER case, although the wind speed in the CBL was stronger than that in the PER case at all downwind positions.

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Vertical profile of the wind speed in the flow direction. The color of the line indicates the distance from the inflow plane. The distance is shown in the figure. PER shows the vertical profile of the horizontal mean (solid line) and ± standard deviation.

Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0053.1

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Figure 4 shows the potential temperature profiles for each downwind location. Because the CBL height is primarily determined by the amount of sensible heat flux from the ground surface, we assumed that a good boundary condition could reproduce a potential temperature profile similar to that of the PER. In the case of the PER, an isothermal layer was observed up to approximately 650 m, including overshooting (transition layer). The original and extended DF methods accurately reproduced the CBL height, although there was a slight downwind overestimation. In the case of the extended R-R method, the CBL height and potential temperature rise were underestimated near the inflow boundary. There are several possible explanations for these results. First, the transition layer at the upper edge of the CBL is not reproduced at the inflow boundary. This error caused the suppression of thermals, particularly near the inflow boundary. Another reason is an error in the scaling constant used to recycle the potential temperature. We assumed a linear relationship between the potential temperature increase and the CBL height in this study because the potential temperature gradient was constant. However, further investigation is required.

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As in Fig. 3, but with potential temperature perturbation.

Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0053.1

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The vertical profiles of the GS-TKE for each generation method of the turbulent components are shown in Fig. 5. We believe that the best method shows a vertical profile similar to that of the PER in all areas. This is because sufficient turbulence is generated from the upwind area of the domain. From this point of view, the profile of the extended R-R method is similar to that of the PER case from upwind compared to the other methods. However, TKE was significantly underestimated at heights above 500 m. This is due to the underestimation of the CBL height mentioned above. The profile was similar to that of the PER case 12 km downwind. In the extended DF method, there was some turbulence at 1 and 2 km downwind. The TKE in the extended DF was larger than that in the original DF 1 and 2 km downwind. The maximum TKE appeared at z = 100 m in the extended DF and was approximately 0.50 m² s⁻² at 1 km downwind. This maximum TKE is weaker and lower in position than that of the PER (0.54 m² s⁻² at z = 260 m). There are two possible reasons for this underestimation. According to Okaze and Mochida (2017a), injected turbulence does not satisfy the conservation laws of momentum and heat. This may have caused the turbulence to decay immediately following the inflow boundary and become weaker than the target turbulence. The second problem is the error in the integral length scale. This error could be a cause of the difference in the shape of the vertical profiles between the PER and extended DF. However, this error was eliminated at 4 km downwind. This indicated that the effect of the length scale on the turbulent component was insignificant when the driver region was sufficiently long. This result is consistent with previous studies (Xie and Castro 2008; Xie et al. 2013).

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As in Fig. 3, but for GS-TKE.

Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0053.1

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The vertical profiles of the vertical heat fluxes of the GSs are shown in Fig. 6. The accuracy characteristics of each method are generally consistent with those described for GS-TKE (Fig. 5). As shown in the GS-TKE, the profile of the extended R-R method was similar to that of the PER. In the case of the original DF, overestimation of the CBL was observed 4 km downwind, and the profile became similar to that of the PER approximately 8 km downwind. In the extended DF method, turbulence was present up to approximately 550 m, and the shape of the vertical profile was similar to that of the PER.

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As in Fig. 3, but for GS vertical heat flux.

Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0053.1

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The horizontal cross sections of the wind speed in the flow direction at z = 180 m (z/z_i = 0.33, where z_i is the estimated CBL height, 546.6 m) and z = 580 m (z/z_i = 1.06) are shown in Figs. 7 and 8, respectively. From these figures, we can confirm the characteristics of each boundary condition as previously described. Considering the results at z = 180 m (Fig. 7), in the case of the extended R-R method, the flow field was similar to that of the PER, even in the upwind area. Turbulence injected from the upwind boundary was observed in the original and extended DF cases. Considering the results at z = 580 m (Fig. 8), both the extended R-R and extended DF methods can generate turbulence, even at this height.

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Horizontal cross section of the wind speed in the main flow direction at z = 180 m (z/z_i = 0.33).

Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0053.1

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As in Fig. 7, but for z = 580 m (z/z_i = 1.06).

Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0053.1

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The horizontal cross sections of the vertical wind speed at the same altitudes as those in Figs. 7 and 8 are shown in Figs. 9 and 10, respectively. First, considering the distribution at z = 180 m (Fig. 9), the extended R-R case has a distribution similar to that of the PER case over the entire domain. The extended R-R method can accurately reproduce not only the horizontal wind but also the spatial structure of thermals, even in upwind areas. In the original and extended DF methods, turbulent flow is injected from the upwind boundary; however, the wavelengths are short, and a certain flow distance (driver region) is required to generate thermals. As shown in Fig. 10, there is a marked difference between the methods. In the case of the extended R-R method, sporadic upwelling due to overshooting was observed but was weaker than that in the case of the PER. This suppression of overshooting was caused by the underestimation of the CBL height at the inflow boundary. In the extended DF method, weak perturbations injected from the upwind boundary were observed. In addition, the extended DF cannot contain thermals in the turbulent components at the inflow boundary. The characteristics of the flow field are similar to those of the original R-R and DF methods. In the case of R-R, a proper turbulent field was generated, but it could not maintain the average vertical profile of the wind speed and potential temperature. This was possible because the vertical profile of the potential temperature was not fixed. In the case of the DF, the proper average vertical profile of the wind speed and potential temperature is maintained; however, it cannot generate a proper turbulent field containing thermals. This is possibly because it uses the mean vertical profiles directly; however, all turbulent information is aggregated at the Reynolds stress and length scales.

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Horizontal cross section of the wind speed in the vertical direction at z = 180 m (z/z_i = 0.33).

Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0053.1

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As in Fig. 9, but for z = 580 m (z/z_i = 1.06).

Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0053.1

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We estimated the power spectral density of the wind speed in the main flow and vertical directions to investigate the accuracy of the turbulence reproduction in more detail. Estimation was performed for the perturbation components in the y direction at 1, 2, 4, 8, 12, and 16 km downwind. In total, 156 grid points were used, excluding 50 points from the lateral boundary in the spanwise direction at z = 180 and 580 m. Power spectra were estimated every 10 s and averaged over a 10-min period.

The power spectral densities of the wind speed in the main flow direction u and the wind speed in the vertical direction w for each method are shown in Figs. 11 and 12, respectively. The spectrum of u at z = 180 m (left column in Fig. 11) shows that the energy was underestimated for wavenumber n > 2.0 × 10⁻³ in the case of the original and extended DF methods. The underestimation of energy was probably caused by the overestimation of the length scale. This underestimation of energy was eliminated rapidly and became unclear 2 km downwind. This is because large-scale turbulence was generated at the inflow boundary, and the energy cascade was properly achieved inside the simulation domain (Keating et al. 2004; Muñoz-Esparza et al. 2015). However, the extended R-R method showed good agreement with the PER case. Looking at the results at z = 580 m (right column in Fig. 11), the energy was underestimated for wavenumber n > 2.0 × 10⁻³ in the case of the original and extended DF methods, similar to z = 180 m. However, the energy was underestimated for all wavenumbers using the extended R-R method. This is because the CBL height was underestimated, and the thermals were suppressed at this height.

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Power spectral density of the wind speed in the main flow direction. The method is shown in the upper-right corner of each panel. The colors indicate the distance from the inflow boundary. The dashed line indicates the result of the PER. (left) At z = 180 m; (right) at z = 580 m.

Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0053.1

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As in Fig. 11, but for the vertical wind speed.

Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0053.1

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Considering the spectrum of w at z = 180 m (left column in Fig. 12), the energy was underestimated at high wavenumbers for the extended DF method. The characteristics of the spectra of the w components were similar to those of the u components. The result at z = 580 m (right column in Fig. 12) also shows characteristics similar to those of the u component at z = 580 m, except that the original DF significantly underestimated the energy 1–2 km downwind. In summary, the extended R-R method is effective at properly generating turbulence, even in the upwind region. The extended DF method can reproduce the CBL height properly but may require a driver region of length > 2 km to generate sufficient turbulence.

To compare the thermal behavior, histograms of the vertical wind speeds at two altitudes, z = 180 m and z = 580 m, are shown in Fig. 13. The samples were the same as those used for the power spectral density measurements. Considering the result at z = 180 m (upper panels of Fig. 13), the histogram has a peak at approximately −0.3 m s⁻¹ in the case of the PER. In addition, the histogram was slightly skewed toward the negative side. These characteristics are similar to those reported in a previous study (e.g., Deardorff and Willis 1985). In the case of the extended R-R method, the histogram has the same characteristics as that of the PER for all downwind positions. The original and extended DF methods showed a peak at approximately 0 m s⁻¹, 1 km downwind. However, the shape of the histogram was similar to that of the PER 2 km downwind. Considering the results at z = 580 m (bottom panels of Fig. 13), the histogram has a peak at approximately 0 m s⁻¹ in the case of the PER. Additionally, the skewness of the histogram was smaller than that at z = 180 m. These characteristics have also been reported by Deardorff and Willis (1985). In the case of the extended R-R method, the frequency around 0 m s⁻¹ was higher than that of the PER at 1, 2, and 4 km downwind. One possible reason for this is that thermals are suppressed at this height. As mentioned previously, the CBL height was underestimated in the extended R-R method. This caused the suppression of thermals, particularly around the top of the CBL.

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Relative histograms of vertical velocity. The method and altitude are shown in the upper-left corner of each panel. The colors indicate the distance from the inflow boundary, and black dashed lines indicate the result of PER.

Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0053.1

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As previously discussed, the length scale of the DF appears to have a significant impact on the accuracy of the generated turbulence components. Therefore, the sensitivity of this length scale was investigated. We use the estimation method of Mellor and Yamada (1974) [Eq. (16), hereafter L_0.1] and Architectural Institute of Japan (2015) [Eq. (17), hereafter L_AIJ]. The L_0.1 is the same method used in this study, except that the coefficient changes from 0.23 to 0.1. The vertical profiles of these length scales are shown in Fig. 14:

$L_{0.1}=0.1\frac{{{\displaystyle \int }}_0^{\infty }q\mathit{zdz}}{{{\displaystyle \int }}_0^{\infty }qdz},\text{\hspace{1em}and}$(16)

$L_{\text{AIJ}}=100{\left(\frac{z}{30}\right)}^{0.5}.$(17)

In the extended DF, the horizontal cross sections of the vertical wind speed for each length scale are shown in Fig. 15. Considering the results for z = 180 m, there is a difference near the inflow boundary. In particular, a turbulence component with a finer spatial scale was generated in the case of L_0.1, which was a shorter scale than the others. The spectrum at z = 180 m (upper panel of Fig. 16) shows that L_0.1 underestimates the long-wavelength component, while L_AIJ and L_0.23 underestimate the short-wavelength components. In the case of L_0.1, the underestimation was not relaxed even 2 km downwind. In contrast, in the case of L_AIJ and L_0.23, the underestimation was relaxed 2 km downwind. This is because energy cascades from long-wavelength components to short-wavelength components. Similar features are observed at z = 580 m (bottom panel of Fig. 16).

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Vertical profiles of length scales. Colors indicate the estimation method of length scale.

Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0053.1

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Horizontal cross section of the wind speed in the vertical direction. The length scale and altitudes are shown in the top-right corner of each panel.

Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0053.1

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Power spectral density of the wind speed in the vertical direction. The length scale and altitude are shown in the upper-right corner of each panel.

Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0053.1

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The performance of the proposed extension varied at different wind speeds. This was because convective cells (rolls) were dominant when the wind speed was low (high). To investigate this, we performed numerical experiments at 2 and 8 m s⁻¹ in addition to 5 m s⁻¹. Based on the initial wind speed and surface roughness parameter z₀ = 1.0 m, the parameter z_i/L (z_i and L are the CBL height and Obukhov length, respectively) at u(200 m) = 2, 5, and 8 m s⁻¹ is z_i/L = 194.7, 10.8, and 2.7, respectively. According to the Christian and Wakimoto (1989), the case of u(200 m) = 2 m s⁻¹ is “random cells only; shear unimportant to cell structure and morphology,” u(200 m) = 5 m s⁻¹ is the case “rolls coexist with random cells but are not necessary for their maintenance; random cells dominate,” and u(200 m) = 8 m s⁻¹ is the case “only roll vortex motion.”

Figures 17 and 18 show the horizontal cross sections of the streamwise and vertical wind speed in the case of u(200 m) = 2 m s⁻¹, respectively. As shown in Figs. 17 and 18, the difference in the inflow turbulence generation methods is unclear. This was because the turbulence generated in the domain was more dominant over the entire region than that injected from the inflow boundary. Figure 19 shows the horizontal cross sections of the streamwise wind speed for u(200 m) = 8 m s⁻¹. In this case, the difference in the inflow turbulence generation method was more pronounced than in the cases of 2 and 5 m s⁻¹. Specifically, in the case of the extended R-R, the region of strong wind speed existed at 5 km and less downwind, whereas the DF existed at 6 km and less downwind. Figure 20 shows the horizontal cross sections of the vertical wind speed for u(200 m) = 8 m s⁻¹. We can observe roll vortices in the extended R-R cases near the inflow boundary, although the random motion of the rolls in the spanwise direction cannot be observed. This indicates that a longer driver (recycling) region is required when the wind speed is higher to reproduce the roll vortices appropriately. In the case of DF, the injected turbulence component was dominant near the inflow boundary instead of the roll vortices. Roll vortices finer than the PER were observed 2–5 km downwind. This region may be the transitional region of turbulence. The cross sections in the upper layer show that the turbulence in the R-R case is generally weaker than that in the PER case, and that in the DF case, the turbulence is weak up to approximately 6 km downwind from the inflow boundary.

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Horizontal cross section of the wind speed in the main flow direction in u(200 m) = 2 m s⁻¹. The inflow turbulence generation methods and altitudes are shown in the top-right corner in each panel.

Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0053.1

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As in Fig. 17, but for wind speed in the vertical direction.

Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0053.1

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As in Fig. 17, but for u(200 m) = 8 m s⁻¹.

Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0053.1

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As in Fig. 19, but for wind speed in the vertical direction.

Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0053.1

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Considering the previous results, the length of the recycling region is a key parameter for generating an appropriate turbulence component in the extended R-R method. Thus, the sensitivity of the recycling region length to turbulence in a thermally driven CBL was investigated. We set the recycling regions with lengths of 2 km, 4 km (initial setting), and 6 km (nx = 50, 100, and 150, respectively) and simulated the thermally driven CBL. The other simulation settings were the same as those described in section 2. Figure 21 shows the horizontal cross section of the vertical wind speed. Roll convection was observed near the inflow boundary in all cases. However, in the case of the recycling region with a length of 2 km, no random movement of the rolls in the span direction was observed. Combining this and the results of the sensitivity experiment on wind speed, the following hypothesis is obtained: As the wind speed increases and roll convection becomes dominant, a longer recycling region should be used to generate the appropriate turbulence component. In particular, the positions of the roll vortices were fixed if the spanwise motion of the rolls could not be reproduced within the recycling region. The spurious periodicity observed in this study has been reported in previous research studies (e.g., Nikitin 2007; Simens et al. 2009).

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Horizontal cross section of the wind speed in the vertical direction using the extended R-R method in different lengths of recycling regions. The length of the driver region and the altitudes are shown in the top-right corner of each panel.

Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0053.1

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4. Conclusions and remarks

In this study, we applied the R-R method to grow thermally driven CBLs. Additionally, we investigated whether the extended DF method could be applied to simulations in which thermally driven CBLs are growing. The results showed that both methods could simulate the features of thermally driven CBLs. Specifically, the extended R-R method could reproduce the turbulence in thermally driven CBLs better than the extended DF method. However, the extended R-R method could not reproduce the height of the CBLs as effectively as the extended DF method. These characteristics are similar to those of the standard R-R and DF methods. Thus, the extended R-R and DF methods can be applied to the simulation of thermally driven CBLs in the same manner as the standard R-R and DF methods without considering atmospheric stratification. The extended R-R and DF methods could be new options for downscaling from meteorological mesoscale to microscale LES models. Sensitivity experiments were conducted on the parameters used in the extended DF and R-R. The results showed that underestimation of the length scale in the extended DF method should be avoided because underestimation of the length scale causes a shortage of large-scale turbulence components. The other point suggested by the results of the sensitivity experiments is that the length of the driver region in the extended R-R method should be sufficient to reproduce the spanwise movement of the roll vortices. It should be noted that the proposed methods are not applicable to the simulation of flow fields with heterogeneous forcing across domain boundaries or when the main flow direction varies with time.

The following issues remain to be addressed. First, the method to estimate the CBL height in the proposed method should be improved. The CBL height, used as a scaling parameter in the extended R-R method, was estimated using only the sensible heat flux. However, it was apparent from the results that the scaling parameter should include a transition layer or be based on prior knowledge of the CBL height. Additionally, the proposed methods are not divergent-free. These issues should be addressed in future studies.

Data availability statement.

Software for this research is described in Ikeda et al. (2015).